Website founded by
MatPlus.Net Forum General Even a frustrated Moriarty might better apply deduction...
You can only view this page!
|(1) Posted by Kevin Begley [Saturday, Mar 24, 2012 18:43]; edited by Kevin Begley [12-03-25]|
Even a frustrated Moriarty might better apply deduction...
There are many excellent articles in Mat Plus 42.
One of the best I've ever read is Odette Vollenweider's Homage (part I) to Andrej Lubosov.
But, I want to comment on Per Olin's article ("The Frustration of Sherlock Holmes"), which deals with a frustration I share, concerning the failure of our Codex to establish "clear-cut [genre] divisions."
I must say, I enjoyed Per's writing style, which frames the issues well, by using a conversation between Watson (the poser of many good questions, representing the reader) and Holmes (the authority, representing Per's own conclusions).
Furthermore, there is good suspense to be found in the building of Holmes' case (at least against the present Codex).
An observant Holmes even notices that the present Codex still fails to define terms as fundamental as, "Orthodox" and "Fairy Chess."
Unfortunately, when Holmes turns to the issue of establishing divisions, he fails to apply the deductive reasoning one would expect, and instead abruptly offers (without supportive logic) several conclusions, which appear quite misguided.
The reader might have benefited from understanding how Holmes (and Per) arrived at these conclusions.
The critical question, which Per has Holmes ask: is there a need to distinguish between Orthodox and Fairies (and Retros)?
Unfortunately, Watson should have asked this question in the article, and Holmes should have provided a direct answer.
Instead, like the Codex itself, Holmes dodges the question, and merely offers a set of suggested divisions, which are largely based upon terms he (himself) does not bother to define!
Suppose Watson had posed the question, in a better form:
1) Is there a need to categorize chess problems? and if so,
2) How can this be best accomplished?
The answer to the first question is elementary: Yes, categorization is necessary.
The proof goes like this...
Nikita M. Plaksin
=1st/2nd Prize, Die Schwalbe, 1971
(= 15+14 )
[I expect everyone knows the solution, but if not, see: http://www.softdecc.com/pdb/search.pdb?expression=probid=%27P0000875%27]
For certain chess players, with a specific interest in solving ONLY problems with a strong relation to the FIDE game (win, draw, #n), an encounter with the above problem may not be desirable (regardless that it may be considered a masterpiece of retro-analysis).
Better yet, imagine you love problems which require retro-analysis, but have no interest in problems related to the FIDE game.
Were it not for clearly established divisions, an encounter with the above problem might require sifting through thousands of "orthodox" problems.
This explains why there is an elementary need for genre divisions.
[aside: as far as I know, Win Chloe does not preserve any indication of a problem's genre, which I often find perplexing. Nevertheless, it is possible to search for such problems, especially by theme (if you know what to look for).]
The second question, is more involved: what's the best way to categorize various kinds of chess problems?
Per's article does not expressly ask this, but it does implicitly propose an answer, in the form of a new arrangement (ortho studies, ortho #n, ortho h#n, ortho s#n, ortho Retro&PG, fairy studies, fairy #n, fairy h#n, fairy s#n, and fairy Retro&PG), plus two new categories:
1) Other Orthodox Problems (a potpourri of everything which obeys Orthodox rules, but is not already covered in Orthodox sub-divisions), and
2) Other Fairy Problems (whatever else remains uncovered).
But, before Watson can even ask the good questions, Holmes runs out of Brandy, and trails off somewhere.
So, let's suppose Moriarty were to then sit down for a follow-up conversation with Watson.
Moriarty: Watson, you must believe me, the person who was drinking brandy with you moments ago could not have been the real Sherlock Holmes.
Watson: What? Impossible! How can I possibly accept this?
Moriarty: I can prove it. I happened to overhear your conversation, and, the suggestions made could only have come from an impostor. Tell me, in the categories proposed, what constitutes the separation of Fairy and Orthodox?
Watson: Yes, I am aware, Holmes skipped out without defining his own terms, despite having railed against this very flaw within the Codex! However, if Holmes were here now, I suspect he would say something like: "it's elementary dear boy -- a fairy problem can be detected by inspection of its elements. If an element (such as the board, aim, stipulation, pieces, conditions, etc) is not according to the standard rules of FIDE Chess, it must be, unorthodox."
Moriarty: Not true. The person who claimed to be Holmes made clear, the Codex (orthodox chess problem rules) is altered over the years, and may continue to be altered. He offered a clear example -- demonstrating that Dead Reckoning may even, as a result of changed orthodoxy in the Codex, result in premature truncation of a solution (in both problems & studies). This might even upend some thematic content!
Watson: Yes, I do believe Holmes, err, the alleged non-Holmes, would therefore insist upon Orthodox problems being categorized according to rules of the present day.
Moriarty: Impossible -- if this were his intention, he would have had no reason to request that the Codex specify whether Chess960 is Fairy or Orthodox. This would be clear, according to modern rules.
Watson: Presuming you accept all of the rules, including the initial setup.
Moriarty: Exactly, Watson. No provision was made to exclude other rules -- why would Holmes require a specific clarification on the initial game array?
Watson: He had a large glass of brandy, and uhh...
Moriarty: Furthermore, Holmes would never have settled for using present day rules to categorize chess problems. He said he desired a "clear-cut division." But, problems today categorized by his suggestion might require re-categorization tomorrow! The divisions he proposes can not possibly be called "clear-cut."
Watson: That was strange, but in the interest of the FIDE game of chess, one is required either to define orthodoxy according to the rules of the present day, or else according to the rules of the publication date.
Moriarty: Exactly, Watson! And consider: the latter method requires only a brief update (dates and changes to the original rules), whereas the former method (which the alleged Holmes prescribes) would require a continual, laborious reclassification of all prior orthodox problems!
Watson: Well, the choice Holmes proposes does seem unsettling. However, your claim is extraordinary, and this is not sufficient proof.
Moriarty: There's more! Consider his additional category: "The Other Problems" (which is in fact two categories).
Watson: That's a terrible name, yes, but it's like a "potpourri" category.
Moriarty: Would you go looking for a book filed under a section titled, "potpourri"?
Watson: I expect not -- thankfully, library science offers a better categorization system than chess problems.
Moriarty: Book genres are far more difficult to classify. Would the real Holmes have failed to recognize that his categorization is grotesque? Yet, he never bothered to express this fact!
Watson: Aren't you assuming a better system is possible?
Moriarty: Precisely! The real Holmes would not have settled for such a poor categorization, unless it was absolutely necessary; yet, no justification was made for it.
Watson: Perhaps he ran out of brandy before he could explain.
Moriarty: And what about the grouping of Proofgames with Retros? Logically, help-games (aka PGs & A->B problems) are more closely related to help-mates -- few require any retro-analysis!
Watson: Yes, but this is where they traditionally go.
Moriarty: A case has already been made for separation, and these branches were divided in a WFCC tournament.
Watson: Holmes may not have noticed this.
Moriarty: The man claiming to be Holmes stated that his interest was in finding "clear-cut divisions" -- no conversation is required to perpetuate tradition.
Watson: But, this doesn't preclude...
Moriarty: What is a retro?
Watson: Any problem which requires an analysis of some moves played prior to the diagram.
Moriarty: Is this a clear division to lump Proofgames (aka Help-games) in with Retros?
Watson: That's an interesting point, which he walked out on. But, where else could he put Proofgames? There are also math problems in the Retro Section -- do they warrant a special category?
Moriarty: Are you sure Math problems remain in the Retro Section? The category name provides no cover for Math Problems -- how do you know they are not grouped into his new, "OTHER" category?
Watson: I see. The folly of his categorization goes well beyond my first impression. It is not even clear what goes where!
Moriarty: And, what about the title components of these divisions?
Watson: You mean, these divisions within the one FIDE album per 3-year period, would impact the point-race towards FIDE titles?
Moriarty: There is not ONE FIDE-Album, there is one book which collates a collection of genre-albums. There are exactly as many albums as there are divisions! Would the real Holmes have ignored the politics of these title implications? Is this not the primary source of tension which prevents progress in categorization?
Watson: It may well be, but Holmes would have no interest in titles -- his purpose, in offering us a proposal for new divisions, was strictly for the benefit of sorting (cataloging).
Moriarty: Ignoring title implications, tell me, what are the fundamental divisions, which you would consider absolutely necessary?
Watson: First and foremost, the Orthodox Chess Game (as defined by the FIDE Rule Book of the era) would merit a special category -- this would constitute the primary division (maybe call this the Orthodox/Fairy division), which must exist for the sake of the game itself. It would include all problems which follow the rules of the game, though the aim/goal/stipulation may be contrary to objectives in the game (providing the stipulation does not conceal a fairy element, which alters or constrains the game rules). The person who claimed to be Holmes had something more or less resembling this...
Moriarty: Not exactly. [correction] He did include series-movers in the fairy division, but unlike other fairy conditions, they are included into a newly proposed "OTHER Fairies" division. Why does one condition (such as ser.h#n) not belong with the others (such as Black Only Moves to Check + h#n)? Where do Parry-Series-Movers go (into the series-division, or into the division with conditions which may result in the identical solution, such as White Must Check)? Holmes would certainly have foreseen this, and provided some insight.
Watson: But, poor stipulation conventions, and a lack of defined terms, have eroded the traditional categories into a fine mess. And, careless new inventors have added to the decay. Holmes can't be expected to fix everything.
Moriarty: The point is, the real Holmes would take care to to define divisions in a consistent manner, across the board. And, he would not have pretended that perpetuating the traditions which caused the mess are a means to resolve it. Even if his intent were to adhere, as much as possible, to traditional divisions, he would have included the established sub-divisions for and #n problems, and the recent (politically questionable) sub-divisions for h#n problems. But, please, ignore the politics, and continue listing the fundamental categories...
Watson: OK, a second sub-division is required for problems which require Retro-analysis. Third, within the Orthodox section, under the non-retro sub-division, one must distinguish between studies, direct-mates, and other problems (for the sake of the chess player audience). These divisions are (arguably) a necessity; but, beyond these, further sub-divisions would be optional.
Moriarty: If that were the real Holmes, and his interest was strictly to define clear-cut categories, why would he have not employed a sub-category structure, based upon these fundamental divisions? There are now two sections for helpmates in the Album, but only one division is recognized for Judges. Similarly, a few FIDE Judges are given license over Math Problems, but (unless I'm mistaken) there is no individual Math genre (it falls into the Retro&Proofgame division, and Retro Judges). So, why recognize a Math distinction for a few judges? Why would Math have a separate distinction, when hs#n problems outnumber them by an overwhelming ratio?
Watson: Yes, all true... and he might have based these sub-divisions upon quantitative problem data from the perspective of both history, and present day (e.g., which sub-genres were popular, and which are currently?).
Moriarty: There is no objective merit established to confer a sub-category distinction -- it's all politics. Composers working within recognized genres (with their own separate FIDE Album, judged by their own internal experts, according to their own internal standards) gain a significant advantage (in the title hunt) over composers working in broader genres (often judged by outsiders). In short, each sub-division spawns greater inequities (along with greater political turmoil). It makes little sense rejecting a problem which narrowly fails in one sub-division, if a successful problem in another genre has lesser objective value. In an effort to balance old failures, Holmes seems to be proposing to add more divisions (it seems unclear whether he intends to merge any existing divisions). While there may be potential merit to an expansion of genres, it may also be short-sighted to narrow the standards further; and, if not done carefully, new genres may offer explosive opportunities for reward. There is a case to be made that the bare minimum number of divisions (read: separate Albums) would result in broader standards, applied more fairly. But, before we start carving out new genres, or merging old ones, an objective sub-division criteria should be established, with an eye focused on providing both concise categorization, and title equity.
Watson: How could Holmes have missed all this? I just don't see how that could possibly have been the real Holmes.
Moriarty: It doesn't matter anyway. Problemists have demonstrated little capacity to get beyond the title-politics of correcting their broken Codex. One frustrated Holmes is as good as another.
|(2) Posted by Kevin Begley [Saturday, Mar 24, 2012 20:13]|
To be perfectly clear, in no way am I calling Per Olin an "impostor."
Quite the contrary -- I respect his composing ability, I appreciate his insights, and I applaud him for taking a courageous stand on this important issue.
I'm simply suggesting that the Sherlock Holmes, found in his article, does not apply an authentic deduction when attempting to resolve this matter.
And, I happen to think that the logic of Holmes would certainly help resolve this issue.
To that end, I hope this advances that purpose.
|(3) Posted by Per Olin [Sunday, Mar 25, 2012 12:52]|
Kevin, thank you for your extensive post and especially for the addition in post 2!
For persons, who have not read the article referred to, it might be difficult to get the whole picture. The article is a conversation that has taken place, statements and counterstatements have been made. There are opinions, there are opposing opinios, there are misunderstandings, there is perhaps the Truth...
It is widely accepted that the categorization of chess problems in the Codex is unsatisfactory. The world around us changes, the chess rules change, the Codex does not change. The purpose of my article has been to put attention to the matter and get some discussion going. We are, hopefully, on that way!
In an article in Springaren ages ago, I proposed an experiment (this is an issue, which is not well suited for a magazine, but a place like MatPlusForum is excellent). We are to start from zero, from a clean table, forgetting all past divisions. Our task is to make a functioning system for classification of all existing chess problems. First we can make two categories, then three, four and so on. I suggest that we here now make this experiment. Kevin and all readers are invited to a fruitful discussion!
To get it started, her my proposal for division into two: 1) Problems based on FIDE rules of chess 2) Problems not based on FIDE rules of chess.
How are we to do this if we have three categories?
|(4) Posted by Kevin Begley [Sunday, Mar 25, 2012 18:30]; edited by Kevin Begley [12-03-25]|
I like the experiment, but we need to first clarify the aim. I presume your aim is to:
1) Simplify the categorization for purposes of categorizing (sorting, searching, dividing, etc).
However, other motivations are, as Holmes would say, "afoot." These motives may include (but are not limited to):
2) Preserving traditional categories (as much as possible), while attempting to improve upon simplification (and perhaps other aims),
3) Improving a distorted title race, through manipulation of the FIDE Album categories,
4) Organizing categories according to personal preference/benefit.
a) for the benefit of (or to prevent harmful impacts upon) one's own title prospects,
b) for more favorable status of personal compositions/inventions, etc...
5) Organizing categories according to a "popular" perception.
When it comes to personal motives, we are all partial, and should consider ourselves suspect.
That said, in the interest of achieving simplification, I would much prefer the Codex start out by defining basic terms (e.g., your suggestions: Orthodox / Fairies).
This would seem easy enough -- problems which obey all of the rules of the "present day" FIDE rulebook for Chess, can be considered Orthodox.
But, this isn't so easy -- there are (at least) two issues:
1) The rules may be tweaked slightly, which (as you well illustrated w/ Dead-Reckoning) may alter categorization (and perhaps even thematic content!) of a problem.
Therefore, Orthodox requires a sub-category (Orthodox according to today's rules, and Orthodox according to the rules on publication date).
2) The traditional divisions would certainly change.
For example, reflexmates are currently grouped with Orthodox problems, despite employing a fairy constraint (which can not possibly be defined as Orthodox).
If we simplification is taken as the primary, paramount aim, I see no reason to divide problems into Fairy / Orthodox.
The purpose of this is primarily a matter of achieving a more homogeneous judgement.
[correction: I take that back -- my above remark is not actually correct. The primary purpose of the Ortho/Fairy division is to achieve a favoritism for standard chess problems (and in fact, only a select subset of those)! The truth is, judging a fairy #n against a fairy h#n is less homogeneous than it would be to separately judge all #n together, and all h#n together. Furthermore, the latter grouping would require judges to consider the ECONOMY OF FAIRY ELEMENTS (rather than carelessly default to the cavalier attitudes we often see).]
I would suggest dividing problems into their primary motive:
1) Opposing- (or forcing)
(including +, =, #n, s#n, r#n, =n, ser.#n, ser-ep-n, pser-#n, etc),
2) Helping- (or scheduling)
(including h#n, hs#n, PGn, A->B, ser-h#n, pser-h=n, h-ep-n, ser-h00-n, etc), and
(including Retros, Math, etc).
* problems involving analysis of moves prior to the diagram (Retros) would be grouped in the Others section (regardless of their apparent stipulation).
Why should the FIDE-variant continue to enjoy a category of special favor?
[Any favor they receive should be based upon their ECONOMY OF FAIRY ELEMENTS, and not upon a biased categorization system.]
Suppose Chess960 (or some other variant, such as an April 1st invention of GM Ivanchuk) were to grow wildly in popularity?
Your categories would then require alteration.
In fact, your categories already require bizarre alterations -- due to traditional inaccuracies, and alterations in the Rule Book (as noted above).
My suggestion, on the other hand, would not vary over time, nor with rule changes.
Therefore, this must be taken as a more simplified form.
And, lastly, it would still be possible to sub-divide these categories (for example, according to the number of moves, or no moves).
This way, you can separate studies/#2/#3/#n (and direct-series-movers would generally fall into the later category).
Of course, some may falsely claim that this does not sort orthodox problems -- but, this is not correct.
First, orthodox can be seen by inspection.
Second, nothing prevents good editors from sorting their columns (according to fairy-economy), such that orthodox problems appear first.
Third, databases have all the necessary mechanisms to filter out fairy board, fairy conditions, fairy pieces, and fairy stipulations (Win Chloe already does this).
Assuming the first division has four-subcategories (null-movers or studies, 2-movers, 3-movers, and n-movers), and the second has two-subcategories (3-ers, and n-ers) this yields seven (7) simple, unbiased, and relatively heterogeneous categories.
*the subdivisions are only suggested to illustrate that it is possible to preserve a number of traditional categories (though it would recast fairy elements as an economic matter, rather than a separate categorization).
A final selling point is: regardless of fashion swings, the above divisions (and whatever their sub-categories) would not experience the drop-offs presently occurring (and expected to increase!) within some traditional orthodox sub-categories.
I highly doubt a more simplified categorization system is possible.
[edit: a possible statistical comparison is possible -- consider the breadth & crossover of composers between your divisions and mine. There are a large number whose work spans one of my divisions with minimal crossover (e.g., they'll do #2, s#2, r#2, fairy #2 etc, but few helpers, or vice versa), whereas I expect a smaller number will work a broad section of the Orthodox/Fairy division, without crossover. The reason for this is fairly evident -- complex themes crossover ortho/fairy boundaries easily; not so much across the opposing/helping divisions.]
All that said, I expect this will encounter considerable difficulty in obtaining a broad acceptance, due entirely to the judgement implications!
|(5) Posted by Nikola Predrag [Sunday, Mar 25, 2012 22:19]|
The long history of chess deserves a greatest respect. Anything that crosses outside the frame of chess rules is not a genuine chess problem. For the sake of the game of chess and millions of chess lovers, a clear cut is mandatory: genuine-chess and quasi-chess problems. Orthodox and Fairy may well remain as the names. Orthodox should include everything that can legally happen in a chess game, even in a most strange one (reflexmates included). So, any chess lover can follow the play, even without understanding the purpose. That is the logic but certainly not a definition. Some minor rules extensions may be considered, from the initial array up even to the chess-time (alternate w/b moves), but within the frame of easy-understanding by chess-lovers.
We will count fairies in chess problems but we can not privatize the game of chess and alter the rules. A clear-cut separation of Orthodox problems is needed to keep us among the chess-lovers. Only those who already know the chess rules can find an orthodox chess problem legible and very few of them might eventually find a fairy problem legible, not to speak of understanding.
|(6) Posted by Kevin Begley [Monday, Mar 26, 2012 02:33]; edited by Kevin Begley [12-03-26]|
On the contrary, respectfully Nikola, it is the above policy which disrespects the very origins of problem chess (Shantraj problems), and perpetuates failed (biased) divisions in the name of a relatively brief tradition.
ALL forms (Shogi, Xiangqi, Korean Chess, Nightriders, Madrasi, etc) are derivative from this origin, and ALL should be considered "genuine" (not only one).
If WFCC fails to recognize universal fairness, it is destined to be enveloped by a larger chess problem community.
I assure you, problemists (and media) would then turn heavily to the alternative.
And, as I've stated, the FIDE variant can easily be filtered by editors/databases, and it may be considered the base, international form (zero fairy elements) by judges.
Chess lovers will have no difficulty finding their preference within these divisions.
If your primary interest is clear divisions, you can offer only worse than my proposal.
Whereas ortho/fairy divisions are a discrimination (on the basis of the present/popular form of the game -- which you presume will not undergo further evolution, as if the addition of Dead Reckoning rules have achieved some perfection), my proposed divisions divide three unique types of PROBLEMS (forcing / scheduling / retro).
Equal rights did not disrespect the majority -- it merely requested a biased system be altered in favor of universal fairness (and mutual respect).
My intent is to simplify problem categories, and to avoid divisions which are based upon shifting sands (such as the present orthodox rules).
The fairy/orthodox division can not be defined, because orthodox has been (since the origin), and may always remain, in a state of flux.
>"We will count fairies in chess problems but we can not privatize the game of chess and alter the rules."
The rules of chess are altered -- historically... even recently, dead-reckoning rules were added, which change the fundamental content of an orthodox problem (conceivably rendering fairy a recent orthodox problem).
You can not legitimately define Orthodox, therefore, you can not divide by it.
Furthermore, if you wish to go by tradition, you have failed to address the bias inherent in reflexmates (which conceal a fairy constraint in their stipulation, yet are divided into Orthodox problems).
|(7) Posted by Kevin Begley [Monday, Mar 26, 2012 05:40]; edited by Kevin Begley [12-03-26]|
Remember, this is an experiment, which aims to simplify problem categories... if there is a more optimal solution than the one I've offered, tell us what it is.
And, to illustrate the folly of attempting an orthodox/fairy division, let's revisit a point raised in Per Olin's article.
Consider the following problem (which was, at the time of publication, an orthodox study):
Lev Loshinsky & Vladimir Korolkov & Alexander Hildebrand
S.Rustaveli JT, 1966
2nd Special prize
(= 9+11 )
1.Sa6 Kb7 2.dxc6+ Kxa6 3.Be6 Qxe6
-(3. ...Qb3 4.Bc4+ Qb5 5.Be2 Qxe2
--(5. ...f5 6.Sxc7+ Bxc7 7.Bxb5+ Kxb5 stalemate =) 6.Sxc7+ Bxc7 stalemate =)
4.Sxc7+ Bxc7 stalemate =
According to thematic content, one might say this nicely shows a few stalemates... and many would agree this makes a nice orthodox study.
In fact, according to FIDE Rules of Chess, only one stalemate position is actually reached (7.Kxb5 stalemate) -- the rest are clipped by Dead Reckoning rules.
The intended content has been affected by changes in orthodox rules.
If this problem was orthodox when published, what is it now?
If we restore it to Orthodox status, by clipping a few stalemates in the intent, then in light of our comments, we can not deny that the content has been somewhat altered.
If we do not restore it, and elect to preserve the intended form, then according to Nikola's argument, this problem was once, but ceased to be, "genuine" -- immediately after the FIDE rules were changed (to incorporate Dead Reckoning).
If in this short span of time, it has decayed into a "non-genuine" element, it can never have been the genuine article.
Orthodox must be a transient, non-fundamental state of problem matter.
If you divide problems into Orthodox/Fairy, you'll have to provide us some definition for your terms "Orthodox" and "Fairy."
And, these definitions will have to account for Dead-Reckoning issues (along with fairy constraints which are concealed in poorly crafted stipulations, such as Reflexmates, series-movers, parry-series-movers, cap-zug, anything using the ##/== aim, etc).
|(8) Posted by seetharaman kalyan [Monday, Mar 26, 2012 05:44]|
Interesting discussion on an essential topic. While generally agreeing with Kevin Begley on the categorisation he suggests 1. Opposing 2. Helping 3. Others, I think there are already Hybrids. Where does Helpselfmates go? helping or opposing. There are also other types like Blundermates, where black helps for some moves and then opposes. Reflexmates are really hybrids where black opposes for some moves and but helps (or made to help by stipulation) in the last move !
|(9) Posted by Kevin Begley [Monday, Mar 26, 2012 05:51]; edited by Kevin Begley [12-03-26]|
I thought carefully about (and already covered!) the question of hybrids.
You sort according to the PRIMARY type of play (that which is used first, not last).
Therefore, hs#n is sorted into helpers -- its primary nature is a scheduling problem, and this is where it belongs.
The secondary nature (s#1), has simply become a new AIM of the problem (like checkmate, or capture is an aim).
The only difference is, the solver is expected to recursively evaluate those aims which constitute a sub-stipulation.
Finally, the true nature of "reflexmates" is an involved discussion.
Some claim there is a "reflexive" fairy condition at work.
I don't believe such a condition is necessary to understand semi-r#n problems (but, this requires a lengthy digression).
Suffice it to say, r#n conceals a fairy constraint, and does not follow the rules of orthodox chess.
Semi-r#n *might* still be considered orthodox, if you consider it to be essentially a direct-problem, with a sub-stipulation in the aim (of "h#0.5-for black."), which requires the solution provide a full recursive evaluation (down to the ultimate aim: white is checkmated).
The point is, we may correct/legitimize stipulations later -- my proposed divisions would be in no way impacted by such changes, since the primary nature of the problem remains intact.
|(10) Posted by Kevin Begley [Monday, Mar 26, 2012 06:34]; edited by Kevin Begley [12-03-26]|
Maybe somebody can offer a better name for my "Others" category; meanwhile, I would suggest changing it to: "Finders."
As in: FIND the game history/illegal cluster/math solution/construction/which player has the move, etc.
Suppose your problem is: you are out of milk, and need more, as does your friend across town.
This merely states an aim -- your primary problem may be altogether different.
You may need to find your car keys, you may need to schedule with your friend, you may need to force your way into traffic.
Milk (for you and your friend) becomes the final aim -- but this aim does not classify the true nature of your immediate problem.
My proposal categorizes problems according to the fundamental nature of the immediate problem.
By contrast, the Orthodox/Fairy categorization attempts to divide your problems according to the make of your vehicle (e.g., Nissan or non-Nissan)!
This fails to adequately confront both the Datsun (analogous to rule changes), and the concealment of a foreign engine beneath a Nissan exterior (analogous to orthodox stipulations, like r#n, which are driven by fairy constraint).
Nobody can provide a logical classification system (nor even definitions for their own terminology!) which follows the traditional, biased form (used today).
Instead, they resort to unsupported claims that the Nissan is a special vehicle, which deserves a unique favoritism in classification.
Even if we should accept this premise, they have not yet provided us a SIMPLE PRIMER for their categorization!
Finally, note that my proposal does NOT preclude journals from excluding problems (based upon a variety of self-identified criteria).
Die Schwalbe is free to limit problems to Orthodox Chess, and its immediate derivative forms (the FIDE Album might even elect to establish such limitations).
The Wenigsteiner-Jahrespreis publication would remain free to restrict entries to Chess derivatives using 4-men or less.
Orthodox-Only publications would remain free to exclude derivative forms of chess.
Furthermore, databases would remain capable of filtering orthodox/fairy problems (to the degree you care to define these terms) -- none of this changes.
I am proposing nothing more than a valid system of categorization, based fundamentally upon the immediate nature of the problem, in the simplest form imaginable.
If there is a simpler alternative proposal, I would be happy to consider endorsing it; but, I highly doubt anyone will provide such a proposal (it is unlikely to exist).
|(11) Posted by Per Olin [Monday, Mar 26, 2012 15:18]|
Some comments (I generally try to express myself briefly and simply; sometimes I manage both, sometimes neither):
- Aim: Kevin mentioned that it is good to specify an aim for this exercise. This could be 1) to make the problem chess community aware of the need to update the classifications of the Codex and 2) as I/we criticize the present system, we have to come up with some better alternative(s)
- Tentatively, we could have two approaches: 1) no need to refer to the chess rules as outlined by Kevin in post 4 and 2) need to keep the division into orthodox and unorthodox/fairy as outlined by Nikola in post 5
- The Kevin approach gives three classes: A) Opposing (forced ) play B) Helpplay C) Others. This reminds much of what feenschach uses. If we consider A and B to be the broad classification, I think there needs to be some subgroups. Here I see some limitations: the total number of classification groups can not be very big, some 8 – 10 might be a good number. Anything that we can not give a subgroup, will have to go into others. Possibilities: Opposing: A1) endgame studies A2) direct mates A3) - Ax Helpplay: B1) Helpmates B2) Proofgames B3 - Bx C) Others. Please, develop further.
- The Nikola approach is the traditional one and comes close to what I have expressed in the article, which has given reason for this thread. Basically, as I see things, there is a call for one more orthodox group, which could be called Other orthodox problems. This would include stalemates, helpstalemates, helpselfmates, reflexmates (!?), mathematical problems, constructional problems if not among retros, etc. etc. Everything else is fairy chess, the dividing line consisting of the chess rules. Yes, they change over time, but that is a matter that will be manageable. We would then have the groups: 1) Direct mates 2) Endgames 3) Helpmates 4) Selfmates 5) Proofgames and retros 6) Other orthodox problems 7) Fairies. If the last group starts to feel very big and heterogeneous, one possibility would be to divide it using the established groups (1a Ortodox direct mates 1b Unorthodox direct mates; 3a Orthodox helpmates 3b Unorthodox helpmates etc.). As far as I can see, this proposal is so close to the present that it might even get acceptance!
- Why have we so far managed without a group Other orthodox problems? Because the group fairies has been available accepting many things that are not fairy, when the definition for fairy is anything where normal chess rules do not apply. - As an exercise, try from the Codex to find out the definition for Fairy Chess!
- - - - - - - - - - - - -
Problems in PDB for the moment:
If we exclude studies, how well do the numbers correspond to the actual situation for problems published nowadays?
|(12) Posted by Kevin Begley [Monday, Mar 26, 2012 16:38]; edited by Kevin Begley [12-03-26]|
First, PDB stats are not reliably representative (of popularity within traditional genre divisions).
For example, it grossly underestimates fairy problems -- mainly because the database has an absolute minimal support for such problems.
Many other factors contribute to the statistical misrepresentation (including the interests of the developers/contributors -- which explains an extremely low number of studies).
Of course, it begs the question -- why would you require a statistical snapshot of genre popularity?
I suspect you may intend to use this data as justification for your subdivisions (which are both arbitrary, and highly political).
I would advise against such folly, and would encourage you to first concentrate on defining your primary divisions...
Second, you requested that I further develop my proposal, with respect to sub-groups -- I have already offered options for subdivisions (in my original post 4).
I did so only to illustrate that my simplified, logical, and problem centered categorization system does preserve a number of options, to filter some traditional categories.
1) Opposing Problems (forced) category might have four subcategories:
a) null-movers (problems not constrained by a move limit: studies, fairy-studies),
b) 2-movers (and under -- #2, s#2, r#2, =2, fairy-#2, etc),
c) 3-movers, (#3, s#3, etc) and
d) n-movers. (and over -- #4+, s#4+, ser-#4+, pser-s#4+, etc), and
2) Helping Problems (scheduling) category might have two subcategories:
a) 2-movers (and under -- h#2, h=2, etc)
b) n-movers (and over -- h#2.5+, h=2.5+, hs#2.5+, PG 2.5+, A->B 2.5+, ser.h#2.5+, pser.h#2.5+, etc)
3) Finding Problems (retros & others) may not require a subdivision (but, further subdivision should be based only upon the number of moves: 0 to n).
All subdivision options can be based entirely upon the number of moves -- it need not resort to arbitrary political decisions, which would require justification by statistical analysis of genre popularity (unlike the subdivisions under the ORTHO/FAIRY proposal)!
Further, I have merely listed some possible subdivision options, which loosely based upon current divisions in the Album.
These are not necessarily suggestions -- I would leave it to the delegates to determine what number of moves best subdivides any of my proposed categories.
Since my subdivisions cross all stipulations, political disruption (fights over title opportunities) would be minimal.
As for the number of categories -- this should yield somewhere (roughly) between 6 and 10; though, you could have as few as only 3.
Third, your proposal still fails to define elementary terms, fails to establish any clear primer (for division), and fails to address its exceptionally poor foundation.
You can not seriously propose this, then carelessly dismiss its fundamental flaws (which you perceive as "negligible").
If the flaws in your proposal are negligible, then by your own reasoning, delegates may equally dismiss (as negligible) all flaws inherent in the present system!
So, your own logic argues against improving a flawed system -- which, obviously, contradicts your very proposal.
If you seriously believe the Ortho/Fairy proposal is valid, you must address its inherent flaws:
1) It divides problems into two major camps (Fairies / Orthodox), while providing no clear definition (nor logical primer) for either.
[simply stated: the O/F proposal is yet to provide any classification.]
2) It discriminates on the basis of something non-fundamental (to problems themselves).
[The O/F division is as arbitrary, and gerrymandered as would be a division based upon problems having Some-Nightriders/No-Nightriders.]
3) It would establish a non-fixed dividing line (orthodoxy is a transient state, requiring reclassification over time).
[This completely undermines the integrity of such a classification.]
4) It would require highly arbitrary sub-divisions (based upon favored stipulations).
[This is inherently unfair, and likely results in continued political dissent (opportunistic title fights).]
5) It fails to achieve the simplification you had aimed for.
[The highest aim of your experiment -- simplification -- proves no better than the present system.]
There are other issues -- but, these are the primary failures of your proposal, which must be addressed, if we are to take you seriously.
I honestly do not believe you will have any luck, however the inevitable frustration of your attempt would help persuade you that my proposal is ideal.
Minor arbitrary tweaks will not fix this broken system -- the scale of the problem calls for a fundamental restoration.
The O/F proposal solves nothing -- sure, it would redraw a few existing lines, but without providing either simplification or integrity.
There is a better way -- my proposal offers real solutions, to the real problems of the current system.
It should be plainly obvious that my proposal offers the simplest, and most logical divisions, based upon fixed features which are fundamental to the nature of problems.
In fact, while we're still waiting for the basic definitions of O/F, mine remains the ONLY proposal.
Further, mine offers good options for subdivisions (which approximate the major traditional genres), which are based upon considerably less arbitrary criteria, and should therefore minimize (or avoid completely) political dissent.
You might expect the entire problem community would be jumping for joy -- if only at my proposal's marvelous simplicity (the experiment's original aim).
But, it is that big cobra which little birds dare not discuss -- the fear associated with potentially altered title opportunities -- which keeps us frozen (in a bad situation).
|(13) Posted by Kevin Begley [Monday, Mar 26, 2012 19:37]; edited by Kevin Begley [12-03-26]|
For comparison, you can see the regularly updated Win Chloe stats table at: http://winchloe.free.fr/newtables.html
In my experience, Win Chloe has far fewer duplicates than PDB (where they are often tagged, but not removed), primarily due to WC's automated duplicate-detection algorithm (which prevents their entry).
note: I don't mean to advertise a commercial product here, and I do not benefit (in any way) from doing so.
But, in the interest of statistical evaluation...
If you want to break down Retros/Fairies further...
# of Orthodox PGs: 4k+
# of problems employing either a fairy condition or a fairy piece: ~52k (some may have retro content, and over 900 are Proofgames).
note: this is not yet all fairies -- depending how you define the term...
# of series-movers NOT employing a fairy condition or fairy piece: ~7k (excludes consequent, but still some may have retro content).
includes: ser-direct(~)n, ser-h(~)n, ser-s(~)n, ser-r(~)n, ser-semiR(~), ser-recipH(~)n
*Parry-series is already correctly identified as a fairy condition, and have already been counted (in the 52k).
# of hs(~)n problems NOT employing a fairy condition or fairy piece: ~500 (some may have retro content).
But, there may still be more fairies yet to be counted...
I have not counted problems which exclusively employ unusual aims (e.g., =n, xn, +n, ++n, h-ep-n, h=n, s=n, etc)...
I have not counted those which employ fairy aims (h##n, ==n, etc).
Also, there are some problems which employ fairy board shapes, etc.
I'm sure I'm forgetting other possibilities...
Suffice it to say, there are probably well over 60k fairies (depending how you define this) out of the 75k (others).
By comparison, PDB probably under-represents fairies by at least a 2:1 ratio (probably higher).
It is difficult to say whether Win Chloe is under-representing these, as well.
Furthermore, Fairies are the most rapidly expanding genre -- and, the speed appears to be accelerating.
Meanwhile, the population of Orthodox is noticeably dwindling.
You'll not find this reflected in overall statistics (you would need to consider a more recent time scale).
It should also be noted that Fairies are, by far, the most encompassing genre.
If Orthodox is a small town, Fairies span a galaxy!
And, maybe nothing better illustrates the folly of the O/F dividing line.
|(14) Posted by Kevin Begley [Tuesday, Mar 27, 2012 01:38]; edited by Kevin Begley [12-03-27]|
Interesting WChloe stats:
No fairy unit/condition: 82k (other fairy aims/stips/boards may be included)
Some fairy unit/condition: 3800
1950 - 1964
No fairy unit/condition: 55k (a few are series-movers, other fairy aims/stips/boards may be included)
Some fairy unit/condition: 3.9k
1965 - 1979
No fairy unit/condition: 37k (2.3k are series-movers, other fairy aims/stips/boards may be included)
Some fairy unit/condition: 7.6k
1980 - 1994
No fairy unit/condition: 71k (1.3k are series-movers, other fairy aims/stips/boards may be included)
Some fairy unit/condition: 10.5k
1994 - 2009
No fairy unit/condition: 110k (4.8k are series-movers, other fairy aims/stips/boards may be included)
Some fairy unit/condition: 24k
Except for a strange dip in Orthodox output (between '65-79), output has increased fairly steadily, to nearly double that of the '50-64 period.
By comparison to the same period, fairy problems have increased more than 7-fold (at least) -- swelling to more than one quarter of total output.
And, there is good reason to believe it is still under-represented, even in this database.
Despite this considerable (and accelerating) output, fairies occupy only a SINGLE genre, compared to SEVEN orthodox genres!
So, we have a major classification failure, and we have a Codex which reads like the instructions for assembling failure...
No wonder you are asking for yet another Orthodox Genre (one suffering from a completely negligible output)!
And, you haven't yet told us what illegitimate fairy stipulations your new "Orthodox" might house (I know it's supposed to be a surprise, but I'm dying to hear the punchline).
I'll tell you why this terrible idea might sell...
Because low yield orthodox sub-divisions are worth more than all the simplicity of real problem classification (and all the integrity of a fair competition) -- that's why!
|(15) Posted by Kevin Begley [Tuesday, Mar 27, 2012 06:40]; edited by Kevin Begley [12-03-27]|
For so long in vain, I have requested numerous problemists to provide me a single concise definition -- it proved beyond all of their greatest abilities.
Finally, you can put down your candles -- the great mystery of our time, has been concluded:
There is no such thing as a Fairy.
Behold what profound depth, what paradox, what Truth has been revealed unto me, literally moments ago, from the mouth of babes, and in the voice of Sean Connery!
There has never been, is not now, and will never be any instance of a Fairy Chess Problem.
Exalted, I cried, "Excellent!"
"Elementary," said the child.
|(16) Posted by Per Olin [Wednesday, Mar 28, 2012 11:39]|
Short comment and short question related to a long copied text:
- A couple of alternatives for classifying chess problems have been presented. This step 2 can have some interest only if we can manage step 1, which is: convince the problem chess community that there is a need to update the Codex.
- A question to all of us concerning the definition of fairy chess in the Codex for Chess Composition (the essential parts copied below). In your opinion, can we from the text conclude what fairy chess is?
- - - - - - - - - - - - - - - - - - - - -
Chapter II - Types of Chess Composition 
Article 5 - Classification according to Stipulations
Chess compositions can be classified into several groups according to their stipulation. Besides the historically developed groups, viz studies, direct mates, selfmates and helpmates , further groups  have developed 
Article 6 - Special Types
Additionally, and independent from the classification according to Article 5, there are a number of special types, including:
(a) Retroanalytical chess compositions
(b) Mathematical chess compositions
(c) Constructional chess compositions.
Article 7 - Classification according to Rules
Furthermore, chess compositions can be classified into those which apply the FIDE-rules of the game of chess  and those which apply modified rules [13,14]
8 Articles 5 to 7 are not intended to be exhaustive. Other classifications are possible and also practised, for example according to the material used (miniature, minimal, Meredith etc.) or according to other criteria.
9 According to this classification, examples of frequently used stipulations are:
(a1) White to move and force a win, without restriction to a specified number of moves (studies).
(a2) White to move and force a draw, without restriction to a specified number of moves (studies).
(b) White to move and mate the black king in a specified maximum number of moves (direct mate).
(c) White to move and force Black to mate the white king in a specified number of moves (selfmate).
(d) Black to move and cooperate with White in order to obtain a mate of the black king in a specified number of moves (helpmate).
10 Further groups are, for example, stalemate or series stipulations etc.
11 Compositions other than studies are usually called problems.
12 Presently, the rules for the game of chess as agreed during the FIDE-congress 1996 in Yerevan are valid. Relevant for compositional chess are Articles 1 to 5.
13 In this context, the terms orthodox, heterodox, fairy and exo are used.
14 Modifications of the FIDE-rules may for example consist in:
(a) Rules (conditions) on which the composition is based (for example maximummer, circe, seriesmover).
(b) Pieces used in the composition (for example nightrider, grasshopper, chinese pieces).
(c) Chess space on which the composition is based (for example chess board with 10x10 squares, cylindrical chess board, multi-dimensional chess boards).
|(17) Posted by Siegfried Hornecker [Wednesday, Mar 28, 2012 14:10]|
Die Schwalbe is free to limit problems to Orthodox Chess, and its immediate derivative forms (the FIDE Album might even elect to establish such limitations).
No, it is not. For other journals this might be true, but for Die Schwalbe it would violate their constitution.
|(18) Posted by Kevin Begley [Wednesday, Mar 28, 2012 17:06]; edited by Kevin Begley [12-03-28]|
Very well, Sigfried, I will reword:
Die Schwalbe's liberties are in no way compromised, by my proposal -- like all other journals, this one may preserve (or amend) their own constitution, at their own discretion.
Does this better suit you?
|(19) Posted by Kevin Begley [Wednesday, Mar 28, 2012 19:37]; edited by Kevin Begley [12-03-28]|
We seem to be in complete agreement on one key point: intelligent Codex modification offers the problem community the possibility of significant benefit.
However, according to your wording, the Codex only requires "updating."
The Codex was recently "updated," to account for changes to the rule book (especially dead reckoning); and, some additional modifications significantly improved a number of issues surrounding retro-analysis.
Your wording begs the question: what change has impacted the Codex, to the degree an "update" has recently become required?
I maintain the Codex has never lived up to its potential, and presently requires many improvements (some of which are fundamental -- such as a definition of terms).
Can we both agree on this?
>"In your opinion, can we from the text conclude what fairy chess is?"
No, Fairy Chess is not presently defined, has never been defined, and there is no reason to expect a proper definition will ever be established.
And, why should it be defined at all -- it does not constitute a legitimate dividing boundary for chess problems.
Nevertheless, this false division has been used to manufacture a fraudulent ratio of FIDE Albums, for the purpose of discrimination, in order that those working under favored divisions might profit handsomely, without competing fairly.
A clear definition is impossible, because its conjugate (orthodox chess) is a transient, and non-elemental property.
Despite a widespread pretense otherwise, FIDE Chess continues to evolve.
When confronted by this fact, any clear definition of fairies appears as a dark shadow creeping over an historically favored subset, which pretends itself "Orthodox."
The Orthodoxians sect can not toss out any previous form of orthodox -- lest the future may cast them out, as well!
Therefore, they must deny the truth: there is no definition for Orthodox or Fairy which satisfies the dishonorable motivation for erecting an artificial Orthodox/Fairy boundary.
The Codex has no duty to establish, maintain, or defend that which would constitute an unfixed, arbitrary, and biased division.
Instead, let's turn out attention to just a few of the many significant oversights in the present Codex.
- - - - - - - - - - - - - - - - - - - - -
>Chapter II - Types of Chess Composition 
>Article 5 - Classification according to Stipulations
>Chess compositions can be classified into several groups according to their stipulation. Besides the historically developed groups, viz studies, direct mates, selfmates >and helpmates , further groups  have developed 
The Codex should entirely rethink Article 5.
The Codex does not define the term "stipulation" anywhere; yet, this Article has arbitrarily divided problems according to stipulation!
With equivalent logic, we might separate problems according to whether (or not) they employ a neutral Nightrider (assuming we have not defined neutrals & Nightriders).
Chess problems should be classified according to WHAT TYPE OF PROBLEM they are -- not according to stipulation, aim, board size, board shape, pieces employed, conditions/constraints employed.
I am presently aware of only three types of problems:
1) Scheduling Problems (to achieve a shared objective, both sides must cooperate, in forward play lasting some number -- possibly indefinite -- of moves).
2) Resistance Problems (to achieve an independent objective, one side must overcome an opposing resistance, in forward play lasting some number -- possibly indefinite -- of moves).
3) Finding Problems (the objective is to discover something, without forward play).
note: problems may have multiple objectives -- and these may be nested in recursive layering -- which depend upon a combination of types.
In these cases, it is advised to classify according to the PRIMARY (immediate) objective.
>Article 6 - Special Types
>Additionally, and independent from the classification according to Article 5, there are a number of special types, including:
>(a) Retroanalytical chess compositions
>(b) Mathematical chess compositions
>(c) Constructional chess compositions.
This requires a fundamental rethinking, as well -- it's not so easy to classify problems according to the "Types" listed.
Some Retroanalysis may be required of EVERY standard chess problem containing castling, or en passant capture (even if only in a failed attempt to discredit an assumed legality of casling).
We certainly would not desire to categorize ALL of these as Retros (Type 6a) -- though many uncritical chess players often profess to desire such a classification!
But, the Codex should rise above such an arbitrary and uncritical error.
>Article 7 - Classification according to Rules
>Furthermore, chess compositions can be classified into those which apply the FIDE-rules of the game of chess  and those which apply modified rules [13,14]
I think we can all agree, this is the weakest Article.
There is no benefit for the Codex to list & defend a few arbitrary choices, among the countless possibilities to divide (or sub-divide) problems.
Instead, they should focus on establishing only the primary divisions of chess problems, which are fundamentally based upon problem characteristics.
Further, they should minimize the politics & bias of any arbitrary sub-divisions deemed necessary, by basing their sub-divisions upon universal characteristics (such as the number of moves -- please recall, this is precisely the sub-divisions which my proposal would establish).
>8 Articles 5 to 7 are not intended to be exhaustive. Other classifications are possible and also practised, for example according to the material used
>(miniature, minimal, Meredith etc.) or according to other criteria.
Articles 5 to 7 serve no fundamental purpose.
They are intended only to codify an arbitrary bias.
They should be repealed.
>9 According to this classification, examples of frequently used stipulations are:
>(a1) White to move and force a win, without restriction to a specified number of moves (studies).
>(a2) White to move and force a draw, without restriction to a specified number of moves (studies).
>(b) White to move and mate the black king in a specified maximum number of moves (direct mate).
>(c) White to move and force Black to mate the white king in a specified number of moves (selfmate).
>(d) Black to move and cooperate with White in order to obtain a mate of the black king in a specified number of moves (helpmate).
>10 Further groups are, for example, stalemate or series stipulations etc.
>11 Compositions other than studies are usually called problems.
>12 Presently, the rules for the game of chess as agreed during the FIDE-congress 1996 in Yerevan are valid. Relevant for compositional chess are Articles 1 to 5.
>13 In this context, the terms orthodox, heterodox, fairy and exo are used.
>14 Modifications of the FIDE-rules may for example consist in:
>(a) Rules (conditions) on which the composition is based (for example maximummer, circe, seriesmover).
>(b) Pieces used in the composition (for example nightrider, grasshopper, chinese pieces).
>(c) Chess space on which the composition is based (for example chess board with 10x10 squares, cylindrical chess board, multi-dimensional chess boards).
All these footnotes serve no fundamental function, other than to codify an implicit favoritism for specific, arbitrarily classified problem subsets.
Nobody needs a Codex to tell them how to apply ridiculous classifications to discriminate against an unfavored subset.
It should replace all the present bias with a concise classification of problem types, based entirely upon the fundamental attributes of problems.
If the Codex is used to suggest an Ortho/Fairy boundary of discrimination, it must answer the real questions concerning this boundary, such as:
* If alternate rules are inherently concealed within a given aim (e.g., "##" and "==" problems, etc), does this constitute Ortho or Fairy?
* If alternate rules are inherently concealed within a given stipulation (e.g., series-movers, parry-series-movers, CapZug, etc), does this constitute Ortho or Fairy?
* If alternate rules are inherently concealed within a given publication date (e.g., published pre-Dead Reckoning, after which intended content has changed), does this constitute Ortho or Fairy?
The content of a problem's character (specifically the economy of fairy elements) are certainly a valid criteria for a discriminating judge.
But, the underhanded bias is not -- and the Album favoritism should be discontinued (at the very least).
These failures are rooted in a poorly crafted Codex... and most, I think, agree these sections need a fundamental rewrite.
However, it must be said that Per's proposed alterations would do nothing to improve or simplify the Codex -- in fact, it would advance the old failures further.
Not only would it maintain the historical favoritisms, it would also arbitrarily expand them into two new preferred categories, despite relatively negligible output.
In spite of an already unjustifiable Album ratio, we are asked to further the gross expanse of Album inequity.
If the Codex does not begin to address the subversion of fairness by an underhanded classification system (or should its pace quicken), it will not be problems which are divided, it will be this community.
|(20) Posted by Per Olin [Saturday, Mar 31, 2012 08:14]|
To Kevin and all others: as things are developing, there is no reason to give solutions to a problem, that officially is not recognized to be a problem; no point in answering questions that have not been asked. There is reason only to ask questions. My question is in post 16.
No more posts
MatPlus.Net Forum General Even a frustrated Moriarty might better apply deduction...